The definitions of a surjection and injection make intuitive sense to me from the etymology of sur and in. Likewise for a bijection when defined as a two-way injection between two sets where the bi comes from Latin for two.
However, the Wikipedia article defines a bijection as:
(That is, the function is both injective and surjective.)
Yes, this is an equivalent definition for a bijection, but was this the intention of the original definition, and is this how it's first presented to students?